The combination of (REF ), (REF ) and (REF ) leads to equation (REF ), equivalently equation (REF ); the derivation is provided in [1]}. Therefore, the soliton gas kinetic equation, equivalently the GHD equation, is nothing else but the Liouville equation for asymptotic coordinates, under the transformation of metric that accounts for the relation between the asymptotic and real spaces induced by the two-body scattering shifts.
